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one /(sqrt(2x+ three)+ one)
1 divide by ( square root of (2x plus 3) plus 1)
one divide by ( square root of (2x plus three) plus one)
1/(√(2x+3)+1)
1/sqrt2x+3+1
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1/(sqrt(2x+3)+1)dx
Similar expressions
1/(sqrt(2x+3)-1)
1/(sqrt(2x-3)+1)

Integral of 1/(sqrt(2x+3)+1) dx

The solution

You have entered [src]

 23                   
  /                   
 |                    
 |         1          
 |  --------------- dx
 |    _________       
 |  \/ 2*x + 3  + 1   
 |                    
/                     
3

$$\int\limits_{3}^{23} \frac{1}{\sqrt{2 x + 3} + 1}\, dx$$

Integral(1/(sqrt(2*x + 3) + 1), (x, 3, 23))

Detail solution

Let $u = \sqrt{2 x + 3}$ .

Then let $du = \frac{dx}{\sqrt{2 x + 3}}$ and substitute $du$ :

$\int \frac{u}{u + 1}\, du$
1. Rewrite the integrand:
  
  $\frac{u}{u + 1} = 1 - \frac{1}{u + 1}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, du = u$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{u + 1}\right)\, du = - \int \frac{1}{u + 1}\, du$
    1. Let $u = u + 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      Now substitute $u$ back in:
      
      $\log{\left(u + 1 \right)}$
    So, the result is: $- \log{\left(u + 1 \right)}$
  The result is: $u - \log{\left(u + 1 \right)}$
Now substitute $u$ back in:

$\sqrt{2 x + 3} - \log{\left(\sqrt{2 x + 3} + 1 \right)}$
Now simplify:

$\sqrt{2 x + 3} - \log{\left(\sqrt{2 x + 3} + 1 \right)}$
Add the constant of integration:

$\sqrt{2 x + 3} - \log{\left(\sqrt{2 x + 3} + 1 \right)}+ \mathrm{constant}$

The answer is:

$\sqrt{2 x + 3} - \log{\left(\sqrt{2 x + 3} + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                           
 |                                                            
 |        1                   _________      /      _________\
 | --------------- dx = C + \/ 2*x + 3  - log\1 + \/ 2*x + 3 /
 |   _________                                                
 | \/ 2*x + 3  + 1                                            
 |                                                            
/

$$\int \frac{1}{\sqrt{2 x + 3} + 1}\, dx = C + \sqrt{2 x + 3} - \log{\left(\sqrt{2 x + 3} + 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4 - log(8) + log(4)

$$- \log{\left(8 \right)} + \log{\left(4 \right)} + 4$$

4 - log(8) + log(4)

$$- \log{\left(8 \right)} + \log{\left(4 \right)} + 4$$

4 - log(8) + log(4)

Numerical answer [src]

3.30685281944005

3.30685281944005

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of 1/(sqrt(2x+3)+1) dx

Limits of integration:

The graph:

Piecewise:

The solution